Random 2-cell embeddings of a given graph G G are obtained by choosing a random local rotation around every vertex. We analyze the expected number of faces, E [ F G ] \mathbb {E}[F_G] , of such an embedding which is equivalent to studying its average genus. So far, tight results are known for two families called monopoles and dipoles. We extend the dipole result to a more general family called multistars, i.e., loopless multigraphs in which there is a vertex incident with all the edges. In particular, we show that the expected number of faces of every multistar with n n nonleaf edges lies in an interval of length 2 / ( n + 1 ) 2/(n+1) centered at the expected number of faces of an n n -edge dipole. This allows us to derive bounds on E [ F G ] \mathbb {E}[F_G] for any given graph G G in terms of vertex degrees. We conjecture that E [ F G ] ≤ O ( n ) \mathbb {E}[F_G]\le O(n) for any simple n n -vertex graph G G .
In a latin square of order n, a near transversal is a collection of n −1 cells which intersects each row, column, and symbol class at most once. A longstanding conjecture of Brualdi, Ryser, and Stein asserts that every latin square possesses a near transversal. We show that this conjecture is true for every latin square that is main class equivalent to the Cayley table of a finite group.
No abstract
The chromatic number of a latin square L, denoted χ(L), is the minimum number of partial transversals needed to cover all of its cells. It has been conjectured that every latin square satisfies χ(L) ≤ |L|+2. If true, this would resolve a longstanding conjecture-commonly attributed to Brualdi-that every latin square has a partial transversal of size |L| − 1. Restricting our attention to Cayley tables of finite groups, we prove two main results. First, we resolve the chromatic number question for Cayley tables of finite Abelian groups: the Cayley table of an Abelian group G has chromatic number |G| or |G| + 2, with the latter case occurring if and only if G has nontrivial cyclic Sylow 2-subgroups. Second, we give an upper bound for the chromatic number of Cayley tables of arbitrary finite groups. For |G| ≥ 3, this improves the best-known general upper bound from 2|G| to 3 2 |G|, while yielding an even stronger result in infinitely many cases.We refer to a partition of a latin square L into k partial transversals as a (proper) k-coloring of L. The chromatic number of L, denoted χ(L), is the minimum k for which L has a k-coloring. As a partial transversal has size at most n, χ(L) ≥ n. On the other hand, we may bound χ(L) from above by applying Brooks' theorem to the graph Γ(L). Proposition 1.1. Let L be a latin square of order n ≥ 3. Thenwith equality holding for the lower bound if and only if L possesses an orthogonal mate.Parts of this work have already appeared in the M.Sc. thesis of the second author [6], which was supervised by the first author. For any undefined terms we refer the reader to [2,9,11]. Although latin square colorings are a natural generalization of the notion of possessing an orthogonal mate, the concept did not appear in the literature until very recently. In 2015, papers introducing the concept were submitted by two separate groups: Cavenagh and Kuhl [3] and Besharati et al.[1] (a group containing the first and third author of the present paper). The two groups independently conjectured the following upper bound.Conjecture 1.2. Let L be a latin square of order n. ThenLatin squares for which this conjecture is tight have been given by Euler [4] in the even case and by Wanless and Webb [15] in the odd case. If true, Conjecture 1.2 is likely difficult to prove, as it implies a pair of long-standing conjectures concerning the existence of large partial transversals in latin squares. These conjectures are attributed to Brualdi and Ryser, respectively. Conjecture 1.3 ([9, 12, 13]). Let L be a latin square of order n. Then (1) L possesses a near transversal, and (2) if n is odd then L possesses a transversal.
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